Upid Peak Resonance

8/12/2019 Upid Peak Resonance

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E.PouiinA. Pomerleau

Indexing terms: PID controller, Nichols chart, Maximum peak resonance specification

Abstract: The paper presents a unified approachfor the design of PID controllers, based on thecontours of the Nichols chart. Using a maximumpeak resonance specification, the controllerparameters are adjusted such that the open-loopfrequency response curve follows thecorresponding contour. This approach gives thepossibility of handling (at the same time) themaximum peak overshoot, the minimum phaseand amplitude margins and the bandwidth of theclosed-loop system. The method can be applied toa wide range of processes and it provides useful apriori information concerning the behaviour ofthe closed-loop system. Different ways areproposed for the calculations of the controllerparameters. An optimisation procedure is firstproposed to illustrate the basic idea of themethod, but the aim of the paper is to providesimple tuning expressions for low-order models.These expressions are given for stable, integratingand unstable processes.

1 IntroductionProportional integral (PI) and proportional integralderivative (PID) controllers are widely used in theprocess industries. The simplicity and the ability ofthese controllers to solve most practical control prob-lems have greatly contributed to this wide acceptance.Many tuning methods using different approaches havebeen proposed in the literature. Most common methodsare based on ultimate cycle information [l, 21, first-order models [3], pole-zero cancellation and internalmodel control [4-71, integral error criteria [S-111 orphase and amplitude margin specifications [12-151.Interesting surveys of these approaches have been pre-sented [16, 171. Comparisons of the performances andthe robustness of well-known tuning methods havebeen proposed [18].

EE, 1997ZEE Proceedings online no. 19971493Paper first received 2nd December 1996 and in revised form 30th June1997E. Poulin is with Breton, Banville et associes s.e.n.c., 325 b o d . RaymondDupuis, Mont-St-Hilaire,QuCbec, Canada J3H 5H6A. Pomerleau is with the DCpartement de Genie Electrique, UniversitCLaval, Sainte-Foy,Quebec, Canada G1K 7P4

In recent papers, different approaches have beentaken to develop new PID tuning methods. An H , con-troller with a PID structure has been proposed by [19].Based on the D-partition, a graphical technique fortuning PID-type controllers has been described [20]. Aconstrained optimisation has been proposed [21]. Thepole placement technique applied to PID control hasbeen discussed [22, 231. Based on a control signal spec-ification and the use of one or two points of the proc-ess frequency response, a frequency-domain designmethod have been presented [24]. A tuning method thatuses a specification on the desired trajectory of theprocess output has been suggested [25]. Finally, PIDtuning formulas based on ultimate cycle informationfor integrating processes has been proposed [26].

Despite the fact that many PI and PID tuning meth-ods are available, the interest in developing new meth-ods can be motivated by the desire to have a moresystematic and general approach and the need for sim-ple and efficient tuning formulas that can be used bynonexperts. The objective of this paper is to present atuning method that uses a unified approach and givesconsistent performances over a wide range of processes.The method is based on the contours of the Nicholschart, and it uses a maximum peak resonance specifica-tion. It is the generalisation of the approach suggestedby Poulin and Pomerleau [27] for integrating andunstable processes. This approach provides useful a pr i -ori information concerning the stability and the per-formances of the closed-loop system and could also beused as an analysis method. Different ways are pro-posed for calculating the controller parameters. Anoptimisation procedure is first proposed to illustratethe basic idea of the method but the aim of the paper isto provide simple tuning expressions for low-ordermodels.2The tuning method is based on the contours of theNichols chart, and the specification is given in terms ofthe maximum peak resonance M , of the closed-loopsystem. The controller parameters are adjusted suchthat the open-loop transfer function (controller proc-ess) Gum = G,(jo)G,(jo) follows the contour corre-sponding to the desired M,. This approach hasinteresting properties. It gives the possibility of simulta-neously handling the maximum peak overshoot Mp,theminimum phase and amplitude margins and the closed-loop bandwidth. The method is general and can beapplied to almost all type of processes. It providesimportant information concerning the stability of the

Basic idea of the tuning meth od

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system and gives the possibility of anticipating itsclosed-loop performances. For this reason, thisapproach could also be used as an analysis tool.Fig. 1presents the frequency response of three open-loop systems on the Nichols chart. The Nichols chartoverlaps the open-loop frequency response G jw) andthe contours that represent constant amplitudes of theclosed-loop frequency response IH(jo)j = IG(jo)l/ 1 +G(jo)l. The systems are composed of a PI controller inseries with a first-order process with delay. The curves(i), (ii) and (iii) correspond, respectively, to a stable, anintegrating and an unstable process. The controllerparameters have been adjusted according to the presentmethod for the following specifications: (i): M, = OdB,(ii): M, = 3dB and (iii): M r = 6dB.

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open-loop phase,degOpen-loop frequency responsesof three types of systemig. 1(i) Stable process, (ii) integrating process, and (iii) unstable process

The main difference between these systems concernsthe phase at low frequency. It is -90 for stable proc-esses, -180 for integrating processes and -270 forunstable processes. Only controllers including an inte-grator are considered to avoid static errors. In the mid-dle frequency region, the GQo) curve follows thespecified contour. To obtain a stable closed-loop sys-tem, the open-loop frequency response curve must passto the right-hand side of the critical point (-180 , OdB).At high frequency, the amplitude and the phase of thesystems decrease and the Gum curve leaves the con-tour and gets further away from the critical point. Theproperties of the approach based on the contours ofthe Nichols chart are discussed.2. IThe maximum peak resonance M , and the maximumpeak overshoot Mp are closely related. For a second-order system, an exact relation exists between M p (in%) and M, (in dB). It is given by [28]:

Overshoot to a setpoint change

1 d1- - ' ] 1'2}Adp = lO0exp{ 7r [1+ J 1 - 10-0.lMrSince the frequency response of most closed-loop sys-tems behaves like a second-order system in the fre-quency region around the resonance frequency [29],eqn. 1 can be used as a guideline for the selection ofM,. Table 1presents interesting values obtained withthis relationship.IEE Pvoc.-Control Theory Appl. , Vol. 144, No. 6 , November 1997

Table 1: Relation between Mr and Mp or a second-ordersystem and min imum stability margins for a contour nlr,

Minimum &, Minimum A,,,(deg.) (dB)r(dB ) Mp( )

0.00 4.32 60.000.25 8.47 58.130.50 10.75 56.330.75 12.73 54.601 oo 14.57 52.932.00 21.17 46.803.00 27.16 41.464.00 32.75 36.78

6.025.905.775.655.545.084.654.25

2.2 Min imum phase and amplitude marginsThe fact that the G Q o ) curve follows, and does notcross, the specified contour ensures that minimumphase and amplitude margins are preserved. The mini-mum phase margin $m corresponds to the difference ofthe phase at which the contour crosses the OdB axisand -1 80 . The minimum amplitude margin Am is givenby the difference between OdB and the amplitude (indB) at which the contour crosses the -180 axis.Table 1 presents the minimum stability margins fordifferent contours M,.2.3 Closed-loop bandwidthThe frequency region over which the GQw) curve mustfollow the specified contour influences the bandwidthof the closed-loop system (ob).he selection of aregion located at high frequency implies that the systemwill have a fast response (or a large bandwidth). Theselection of the frequency region is discussed in Section3.2.

The tuning method based on a M , specification thusgives the possibility of handling, at the same time, themaximum peak overshoot, the minimum stability mar-gins and the closed-loop bandwidth. As a firstapproach in calculating the controller parameters, thesearch is formulated as an optimisation problem. Thedistance between the open-loop transfer function andthe specified contour is directly minimised over a fre-quency band. Despite the optimality of this solution,efforts will be directed to develop simple approximaterelations. The optimisation procedure can become acomplex procedure for some systems or controllers.Moreover, the acceptance of a tuning method and itsutilisation in the process industries principally rely onits ease of use [24]. Simple relations based on the opti-mal formulation of the approach will be established. Inthe present case, the formulas are limited to processesfrequently encountered in the industries, (i.e. stable(aperiodic), integrating and unstable processes), eventhough the method can be applied to a wider range ofsystems (e.g. complex poles systems).3The optimal form of the tuning method consists of thedirect minimisation of the distance between G jo ) andthe contour M, over a frequency region. This is a gen-eral approach that can be used for high-order processesand for any transfer function-type controllers. The dis-tance between GQm) and the contour M , is calculatedin the Nyquist plane for mathematical convenience. Anelliptic contour on the Nichols chart becomes circular

Optimal form of the tuning method

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in the Nyquist plane [30]. Let h be a particular contourand X(m) and Y(m)be the real part and the imaginarypart of G jm). The equation of the contour is:

(2)J X 2 ( W ) Y w )J X w ) 1 2 Y U )When h = 1, the equation of the contour in the Nyquistplane is a straight line parallel to the imaginary axis:

X ( w ) = -1 / 2 3)and the distance, at a particular frequency mi, etweenGGm) and the contour is given by:

di I X ( W ~ ) 1/21 h = 1 (4)When h > 1, the contours are circular:

X W ) h 2 / ( 1 h2))2 Y u) = ( h / ( l h 2 2(5)

and the distance at a particular frequency is:hdi = / ( X ( u i ) 2 ) 2 + Y 2 ( w ; )+h2 1 h2

3. I Optimisation problem formulationThe problem of finding controller parameters such thatthe distance between Gum) and Mr is a minimum overa frequency range is formulated as a constrained opti-misation problem in the frequency domain. Using eqns.4 and 6, the minimised criterion is given by:J ( Q c ) 5 X(WZ) + 1/21 M T = OdBa = l

r .

M T > OdB(7 )

where 0, represents the controller parameters. Con-straints are introduced to preserve some properties ofthe system. These constraints are:20 log IH( jw , ) = M T 8 4IH PJ)l2 1 w 5 U T ( 8 b )LG(ju,..) > -180 (84

where wr is the closed-loop resonance frequency andw,, is the open-loop Crossover frequency. The first con-straint (eqn. Sa) ensures that the specification is metand not exceeded. The G(jw) curve follows but does notcross the contour. The second constraint (eqn. 8b)ensures that the relationship between M, and M p is pre-served. The closed-loop system must satisfy this rela-tion to behave like a second-order system in the middlefrequency region [29]. Moreover, when this constraintis not respected, the undershoot can be as important asthe peak overshoot. This could considerably increasethe time response. Finally, the last constraint (eqn. 8c)568

ensures that the system is stable (i.e. that the frequencyresponse curve passes to the right-hand side of the crit-ical point).3.2 Selection of the frequency rangeThe criterion (eqn. 7) is minimised over a frequencyband. This gives the possibility of emphasising ordepressing a particular frequency region. This Sectionproposes guidelines to select proper frequency rangesand gives the ones used in this paper. However, it ispossible to use different regions to obtain systems withdifferent dynamic properties. Processes are gatheredinto three groups: stable (aperiodic), integrating andunstable. Processes with complex poles are omittedhere because they are rarely encountered in processindustries even though the method is applicable.For stable processes, the frequency region where it isimportant to follow the specified contour is locatedbetween mco and f q 8 0 mlg0 is the frequency at whichLGGw) = -180 ). At lower frequencies than coco, theGQm) curve almost follows the OdB contour and aboveq 8 0 , it is desirable to let GGm) go further away fromthe specified contour for robustness and noiseimmunity considerations. For processes with /GO..) OdB. Fig. 2 presents a contour in the Nyquist plane.The contour is a circle centered at 0 (h2/(1 h2), 0)with a radius:R = h / (h2 1) 9)

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The right-most points of the contours on the Nicholschart are located on the straight line X = -1 in theNyquist plane. According to Fig. 2, w1 nd wk re cho-sen such that GGu) is intercepted by the prolongationof the vectors OA and OB. This gives a frequencyregion centred around the right-most point of the con-tour (in the Nichols plane) and approximately centredaround CO, The value of the angle 9 Fig. 2) is givenby:where cp = arccos(L/R) (10)

(11)= h2/ h2 1) 1The parameter or is usually chosen between 0 and 0.9.In this paper it is taken as cr = 0.75. As stableprocesses, the limiting (and theoretical) case of LG jw)-+ -90 is not considered.4 Approximate f o rms of the tuning methodThe previous Section has presented the optimal form ofthe tuning method. Even if there exist powerful tools toeasily resolve constrained optimisation problems,simple analytical relations to calculate the controllerparameters are still attractive. Simplicity and ease ofuse are important factors that contribute to theacceptance of the tuning method [24]. This fact is wellillustrated by the Ziegler-Nichols rules. They are stillused and discussed even if more powerful methods areavailable. Moreover, simple relations are well suited forautotuners and adaptive controllers. Finally, therelations can also be used to initiate the optimisationalgorithm.This Section presents simple tuning formulas basedon the contour concepts for PI controllers with the fol-lowing transfer function:

K,(1 Tis)Tis, ( s ) =The approximate expressions are given for stable, inte-grating and unstable second-order models with timedelay. These types of model can be used to conven-iently represent most types of industrial processes [8].4.1 PI tuning for stable processesThe formulas are established for a second-order modelwith a time delay including a positive zero (nonmini-mum phase). The model transfer function is given by:

T I > T ~ > O i O T o 2 0 13)The negative zero case is not discussed since this zerocan be cancelled. The basic idea of the approximatemethod is to use Tio properly shape Gum in order tofollow an average contour, and to use K, to bringG ( j o ) to the specified contour. This method assumesthat w,,= wr (i.e. that G(jw) is tangent to the specifiedcontour near CO, or the evaluation of K,).The adjustment of the integral time constant dependson To,Tl , T2and 0, i.e.The determination offo(To, Tl, T2,0) using the optimalprocedure is a complex task and some simplificationsare proposed. The integral time constant is assumed,after a normalisation with respect to T I , o have the

T, = fO(TO,~llT21Q) 14)

IEE Proc-Control Theory Appl . , Vol. 144, No. 6, November 1997

following form:T% 1+ f Q/TI)+ g(T2/Z)+ h(TO/Tl))T (15)

This means that the resulting T, is obtained by addingthe contributions of 8, T2 and To to TI . Eqn. 15assumes that there is no interaction between 0, T2 andTo. The functions f (8/Tl) ,g(T2/T1) nd h(To/TJ areestimated by fitting low-order polynomials to the val-ues obtained with the optimal procedure using Mr =0.25dB. This specification is used since Mr = 0.25dBapproximately leads to an 8 to 9% overshoot. This cor-responds to the average accepted overshoot (0 to 15%)for this type of system in process industries. Moreover,the evaluation of Ti is fairly independent of M, sincethe contours have a similar shape below the OdB axison the Nichols chart. The estimated functions aregiven, respectively, by:

@ / T iFig.3 Optimal f(O/TI) and approximatef(O/TI)

T2 /TIFig.4 Optimal g(TJT,) and approximate grTiT,)

Figs. 3 and 4 present the optimal and the estimatedfunctions. According to experimental conclusions, theeffect of To on Ti an easily be neglected, thus,

h(To/T1)= 0 (18)569


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The equation of integral time constant is given by:( l + 0 . 1 7 5 $ + 0 . 3 ( ~ ) 2 + 0 . 2 ~ ) I $ 5 2

Tz= { (0.65+0.35&+0.3(2)2+0.22) Ti E.> 2. \ 19)Afterwards, knowing that G jm) has the appropriateshape (GOB) follows the average contour M , =0.25dB), the proportional gain K, is adjusted such thatGum) is tangential to the specified contour at coco. Thedesired phase at mc0 corresponds to the angle of theintersection point of the specified contour and the OdBaxis at the right-hand side of the point (OdB, -180) onthe Nichols Chart (see Fig. 1). The desired phase isgiven by:

20)and the crossover frequency wco s obtained by solving:

= 7r/2 + arctan(T;w,,) arctan(Towco)4 = arccos (1 1 0 - ~ . ~ ~ / 2 ) T

arctan(TIw,,) rctan(?iw,,) Qwco21)

The proportional gain is then given by:TiK p

(T1T2)2w,60 (T?+ T , 2 ) W & wZoK - - { (TZTO)~W,~,(T,+ T,~)w%1no \

4.2 PI tuning for integrating processesApproximate relations are obtained for a second-orderintegrating model with time delay. The transfer func-tion of the model is:

Section 3.2 has described the selection of the frequencyrange for the optimisation. For integrating processes,the frequency band is centred on the right-most pointof the contour on the Nichols chart. To obtain an ana-lytical solution, this region is reduced to only one pointq, 0). In simple terms, the controller parameters areadjusted such that the point where the phase of theopen-loop transfer function is maximum G jw,,J islocated at the right-most point of the specified contour.The co-ordinate of this point (A,,, $Jmux)are given by:

4 m a z arccos ~ ~ / 1 0 ~ 0 5 M T )T (25)This approach leads to a set of three equations withthree unknowns (Kc,Ti and coma,):

where

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It is worth noting that, for this type of processes, theclosed-loop transfer function has an important zerolocated at -1/T, since T, is generally greater than Tl.This leads to large overshoots to setpoint changes. Thephenomenon can also be explained by the maximumpeak resonance that is typically high (3 to 6dB) forintegrating processes. The selection of Mr for integrat-ing processes is discussed by [27]. An efficient way toreduce the overshoot is to applied a first-order filter tothe setpoint. The time constant of the filter is adjustedto cancel the zero of the closed-loop transfer junction(i.e. Tsp = T,).4.3 PI tuning for unstable processesThe transfer function of the second-order unstablemodel with time delay is given by:

K p - e TI > 0 (30)(1 T S) l T2s)P ( 4 =The same approach as the one used for integratingprocesses is taken to establish approximate relations.The controller parameters are adjusted such thatGGcL),~~)s located on the right-most point of the speci-fied contour a,, 0). For this type of system, the abil-ity of PI controllers to stabilise the system is limited.The following relation must be satisfied to respect theNyquist theorem:

L G j w m a z )= arctan(T,wmax)+ arctan(Tlumax)arctan(T2wmax) Qw,,, 37r/2

Since the PI controller always reduces LGUm), eqn. 31can be reduced to:arctan(TIUma,) arctan(T2wmax) Qwmaz > 0

Moreover, the specification must be chosen such that:

> 7r 31)

( 3 2 )axctan T~wmaz)-arctan T~wmax)-Qwmax7-dmaz

This inequality means that the phase of KcGp(jm) atCO must be greater than $Jmu,. When eqn. 33 is notrespected, the PI controller cannot bring GO@) out ofthe specified contour since it always reduces the phaseof the open-loop system. When the controller can stabi-lise the system, the frequency mmux s located between 11Tl and UT2.Solving the set of equations given by eqn. 26 leads to[27]:

( 3 3 )

Tz = 4Tl(Q+ T 2 / (Ti(4max + ~ / 2 ) ~4Q+ T))(34)and

TtA m a xK , = ~ KP

wherewl = d(T1+ Tz)/ Tl Tz @ + T2)) 36)

As integrating processes, the specified M, are typicallyhigh [27] and the closed-loop transfer function has alarge zero. A first-order filter with Tsp = T,can be usedto reduce overshoots for setpoint changes.IEE Proc -Control Theory A p p l , Vol. 144, No 6, November 1997


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4.4 PID tuning for stable, integrating andunstable processesThe approximate relations for PID tuning are based onthe ones obtained for the calculations of the parametersof PI controllers. The derivative time constant Td isused to cancel T2 (stable and unstable processes) or T I(integrating processes). The open-loop transfer functionis then given by:K,(T,s l) TdS + 1)G ( s )= G, (SI

TZS TfS 1)

( 3 7 )Then, the tuning rules presented for PI controllers canbe applied using the model GJs). For stable, integrat-ing and unstable processes these models are given,respectively, by:

KPe-G ; ( s ) = s 1 TfS)and

Ti > 0 (40)KPecBsGL s) = (1 TIS) l+ TfS)Except for the cancellation of a stable zero (Tf To),the filter time constant is selected as a fraction of Td(i.e. Tr = aTd).The selection of a relies on the noiselevel and the desired acceleration of the system. Thisparameter is typically chosen between 1/3 and 1/10 [31].It is important to note that for unstable processes, theselection of a influences the ability of the controller tostabilise the system (see eqn. 31).5 Examples and resul tsThis Section presents an evaluation of the tuning meth-ods previously discussed through different examples.The first example puts in evidence interesting propertiesof the tuning method based on the contours of theNichols chart. The second example discusses the accu-racy of the approximate tuning relations compared tothe settings obtained with the optimal procedure.Finally, the last example compares the performances ofthe proposed method with those obtained by com-monly used methods. In this paper, only stable proc-esses are considered for the comparisons. Theevaluation and the comparison of the method for inte-grating and unstable processes have been presented[27]. The approximate method has been compared tothose suggested in [32, 331 and has given generally bet-ter results.5.1An interesting property of the proposed tuning methodis the ability to handle the overshoot via the maximumpeak resonance specification. The relation between Mpand M , (eqn. 1) remains an acceptable guideline for awide range of processes. This property is illustrated fora first-order process with delay, since this model can beused to represent many industrial processes. The trans-

Properties of the tuning method

IEE Pro,.-Control Theory Appl., Vol. 144, No. 6, ovember 1997

fer function of the process is given by:% ( S ) = (41)

Table 2 presents the PI parameters obtained using theoptimal approach and Mp for different delays 13.Thespecified maximum peak resonance is 0.25dB. Theresulting overshoots are about 9% as predicted byeqn. 1. The overshoots are almost constant for a )/TIratio going from 0.2 to 5. Fig. 5 presents the setpointchange response of the different systems. The tuningmethod gives consistent performances over a widerange of )/TI.Table 2: PI settings and max imum peak overshoots forthe first-order process with different delays given byeqn. 41 CMr 0.25dB8 S) M,( ) K, T; s )0.2 9.2 2.82 1.020.5 8.9 1.18 1.051 9.0 0.67 1.142 9.2 0.44 1.405 9.2 0.33 2.38

1.2

t im e , sFi 5de given fy eqn. 41A4 = 0.25dB

Set oint change responses for jrst-order process with dzerent

The use of the contours of the Nichols chart alsogives the possibility of anticipating the behaviour of aclosed-loop system. The suggested approach could beused as an analysis tool. Consider the integrating proc-ess given by:(42)e-'Gp(s)=

The PI parameters for a specification M , = 4dBobtained by optimisation are K, = 0.47 and Ti = 4.71s.The open-loop transfer function (-) of the system ispresented in Fig. 6.The transfer functions of systemswith different proportional gains (Tis fixed at 4.71s)are also presented (K, = 0.10, 0.27 and 0.80). The cor-responding setpoint change responses are shown inFig. 7. The interesting point is that increasing ordecreasing the proportional gain for this type of proc-ess results in a less stable and a less damped system. Inboth cases, the GO ) curve crosses more concentriccontours and passes closer to the critical point (Fig. 6).The idea of following a contour is thus an efficient wayto consider the stability and the performance of the57 1


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closed-loop system. This point is particularly evidentfor the systems with proportional gains of 0.47 and0.27. Both systems have the same phase margin (38.1 )but the second one has a larger amplitude margin(14.3dB compared to 9.5dB). Even if the second systemhas larger stability margins, it is less stable and lessdamped than the system designed to follow the 4dBcontour.

1 .6 -

1.2-

open-loop phase, degFig.6 Open-loop i e uency responses G(jw) for dserent proportionulgains T, ixed at 4 . 7 1 ~ 1o r process given by eqn. 42

~ K, = 0.47K, = 0.27_ _ -. . K, = 0.10K , = 0.80r I

0 10 20 30 LO 50time,sFig.7fixed at 4.71s) for process given by eqn. 42Setpoint change responses for dserent proportional gains (T,

K, = 0.27K, = 0.10-. K, = 0.80_ _ _ K, = 0.47_ _ _ -....... . . .

5.2 Comparison of the optimal and theapproximate tuning methodsThis example presents the results obtained with theoptimal approach and the approximate tuning method

for four different types of process. These processes arenamed A, B, C and D and they are given, respectively,by:e-2s

G p a ( S ) = (43)14 s 2(44)(1 s)ePsG p b ( S ) = (1+ s ) ( l+ 0.5s)

e-0 .5sG p c ( S ) = s(1 s)

and(45)

The specifications and the tuning parameters are givenin Table 3. Figs. 8 to 11 present the response of thedifferent systems to a setpoint change followed by astep disturbance applied at the process input. Gener-ally, both optimal and approximate responses are simi-lar. For process A (Fig. S , the response obtained withthe approximate method is a little bit faster than theresponse obtained with the optimal approach. It isworth noting that the step disturbance (applied at theprocess input) has an important effect on the output ofthe system since the delay 8 is two times greater thanthe dominant time constant T I . The responses pre-sented for process B (Fig. 9) are practically identical.For the integrating (process C, Fig. 10)and the unsta-ble (process D, Fig. 11)processes, the responses areclose to one another. For these processes, a supplemen-tary response is presented. It corresponds to the opti-mal approach when a first-order filter with Tsp= Tisapplied to the setpoint. This filter mostly eliminates theovershoot.

Fig.8 Process A ( e n 43) : responses to set point changefollowed bystep disturbance applie%to process inputptimal methodapproximate method- - _

Table 3: Optimal and approximate PI settings for processes A, B,Cand D, respectively, given by eqns. 43 t o 46

Process A Process B Process C Process Dn/l,CdB)Method 0 4 8

Opt. Approx. Opt. Approx. Opt. Approx. Opt. Approx.Kc 0.37 0.42 0.31 0.31 0.34 0.36 3.22 3.26T4s) 1.71 1.85 1.33 1.35 6.94 7.61 1.36 - 1.46

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I I

1.2 n J

-0.210 10 20 30 40 60t i me , sFig.9 Process B y4): responses to set point change followed h jstep disturbance applie to pvocess input

~ optimal method~~~~ approximate method

methods considered for the comparisons are the ITAEmethod [I81 for setpoint tracking (ITAE-setpoint) anddisturbance rejection (ITAE-load) and the pole-zerocancellation (PZC) method [17]. To obtain fair compar-isons, the proportional gain for the PZC was adjustedto produce the same Mp as the one obtained with theATMC. The PI parameters, the maximum peak over-shoots, the stability margins, the 95% rise time (t,) andthe +5% settling time ( tJ are presented by Table 4.Fig. 12 presents the response of the different systemsto a setpoint change followed by a step disturbanceapplied at the process input.Table 4: PI settings and stability/performance indicatorsfor the comparisons of different tuning methods forprocess A given by eqn. 43MethodTMC 0.42 1.85 7.2 60.0 8.3 6.9 10.9

PZC 0.18 1.00 7.2 59.2 10.5 9.5 16.2ITAE-setpoint 0.47 2.13 4.1 63.2 8.1 6.8 6.8ITAE-load 0.55 2.23 10.9 59.9 6.9 6.0 13.8

t ime,sFig. 10step disturban ce applied to process input

~ optimal method~ _ _ _ pproximate methodProcess C (eqn. 45): reFonses to setpoint changefollowedby

........... optimal method with setpoint filtert i me . s

0 2 L 6 8 10t i m e , sFig. 11step disturbance applied io process input

~ optimal method_ _ ~ ~pproximate methodProcess D (eqn. 46 : responses io set point change followed by

.......... optimal method with setpoint filter

5.3 Comparisons to other tuning methodsThe performances o f the approximate tuning methodbased on the contours (ATMC) are compared to thoseobtained with commonly used methods. The compari-sons are presented for process A (eqn. 43). The tuningIEE Proc-Control Theory Appl., Vol. 144, No. 6, November 1997

Fig.12applied to process input (eqn . 43)~ ATMC

Responses to set point change followed hy step disturbancePZC_ ~ ~.......... ITAE-setpoint

~. ITAE-load

For the same overshoot, The ATMC has a shorterrise time and settling time than the PZC method. Theload disturbance response is also faster. The ATMCthus gives generally better results than the PZC methodfor both setpoint tracking and disturbance rejection.Compared with the ITAE-setpoint method, the ATMCapproximately leads to the same rise time but gives alarger overshoot. The settling time for ITAE-setpoint isshorter than the one obtained with the ATMC. TheITAE-load method presents larger overshoot andundershoot. The undershoot increases the settling time.The load disturbance response is faster than ATMCafter the application of the disturbance but both meth-ods bring the process output to the setpoint in approx-imately the same time. The use of the ATMC seems tobe an interesting compromise when setpoint trackingand disturbance rejection are considered. Moreover,the Mr specification could be selected to obtain betterperformance for a particular application. The ITAEmethod offers no flexibility concerning the specifica-tion.

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6 Conc lus ionsThe paper has presented a unified approach for thedesign of PI and PID controllers. The controllerparameters are adjusted such that the open-loop fre-quency response (controller process) follows a contourcorresponding to the desired maximum peak resonance.This method has been shown to be an efficient way tohandle the maximum peak overshoot, the minimumstability margins and the bandwidth of the closed-loopsystem. Examples have illustrated the use of thisapproach to interpret and anticipate the behaviour ofclosed-loop systems. The calculations of the controllerparameters have first been formulated as an optimisa-tion problem. Simple tuning formulas has been devel-opped to facilitate the use of the method. Theseexpressions have been given for second-order modelswith delay since they can be used to represent mostindustrial processes.Different examples have been presented to comparethe performances obtained with the optimal approachand the approximate formulas. The approximate rela-tions have given consistent performances over a widerange of processes and results similar to the optimalones. The proposed method has also been compared toother commonly used methods (ITAE and pole-zerocancellation) for stable processes. The resulting per-formances were generally better than pole-zero cancel-lation, but similar to the performances obtained withthe ITAE method. The main advantages of the pro-posed method are the generality of the approach, the apriori information provided by the method, the possi-bility of using different specifications and the use ofsimple relations instead of optimisation procedures.Comparisons of the method to those proposed by pre-vious worker for integrating and unstable processes hasbeen Dresented in r271.

ReferencesZIEGLER, J.G., and NICHOLS, N.B.: Optimum settings forautomatic controllers, Trans. ASME, 1942, 64, pp. 759-768HANG, C.C., ASTROM, K.J., and HO, W.K.: Refinements ofthe Ziegler-Nichols tuning formula, ZEE Proc. D, 991, 138, pp.COHEN. G.H.. and COON. G.A.: Theoretical consideration of11 1-1 18retarded control, T r a n s A S M E , 1953,75, pp 827-834RIVERA, D E , MORARI, M , and SKOGESTAD, S Inter-nal model control 4 PID controller design, Znd Enn Chem. ~Process Des. Dev., 1986, 25, pp. 252-265MORARI. M.. and ZAFIRIOU. E.: Robust nrocess control(Prentice Hall, Englewood Cliffs, N.J., 1989)FRUEHAUF, P.S., CHIEN, I.L., and LAURITSEN, M.D.:Simplified IMC-PID tuning rules, ZSA Trans., 1994, 33, pp. 43-59RIVERA, D.E., and GAIKWAD, S.V.: Digital PID controllerdesign using ARX estimation, Computers Chem. Eng., 1996, 20,pp. 1317-1334

8 SEBORG, D.E., EDGAR, T.F., and MELLICHAMP, D.A.:Process dynamics and control (John Wiley, New York, 1989)9 CHAO, Y.-C., LIN, H.-S., GUU, Y.-W., and CHANG, Y.-H.:Optimal tuning of a practical PID controller for second orderprocesses with delay, J . Chin. I. Ch. E., 1989, 20 pp. 7-1510 ZHUANG, M., and ATHERTON, D.P.: Automatic tuning ofoptimum PID controllers, ZEE Proc. D , 1993, 140, pp. 216-22411 SUNG, S.W., and LEE, I.-E.: Limitations and countermeasuresof PID controllers, Znd. Eng. Chem. Res., 1996, 35, pp. 2596-261012 ASTROM, K.J., and HAGGLUND, T. : Automatic tuning ofsimple regulators with specifications on phase and amplitude mar-gins, Automatica, 1984, 20, pp. 645-65113 HAGGLUND, T., and ASTROM, K.J.: Industrial adaptive con-trollers based on frequency response techniques, Automatica,14 HO, W.K., HANG, C.C., and CAO, L.S.: Tuning of PID con-trollers based on gain and phase margin specifications, Automat -ica, 1995, 31, pp. 497-50215 TAN, K.K., LEE, T.H., and WANG, Q.G.: Enhanced auto-matic tuning procedure for process control of PUPID controllers,AZChE J., 1996, 42, pp. 2555-256216 KOIVO, H.N., and TANTTU, J.T.: Tuning of PID controllers:survey of SISO and MIMO techniques. Proceedings of the IFAC

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Upid Peak Resonance